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arxiv: 2605.18394 · v1 · pith:AHV5FCPSnew · submitted 2026-05-18 · 🪐 quant-ph · cond-mat.other

Topologically protected long-range correlations in steady states of driven-dissipative bosonic chains

Miguel Clavero Rubio , Tom\'as Ramos , Diego Porras This is my paper

Pith reviewed 2026-05-20 11:04 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.other
keywords non-Hermitian topologydriven-dissipative systemsbosonic correlationssteady-state correlationslong-range orderquadratic Liouvillianssingular value decompositionGaussian decay
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The pith

Non-Hermitian topology appears directly in the long-range correlations of steady states in driven-dissipative bosonic chains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a framework that connects topological phases of driven-dissipative bosonic systems to their steady-state correlations through the singular value decomposition. It claims that non-Hermitian topology in quadratic Liouvillians is encoded intrinsically in these correlations, which display long-range order at fixed frequency and Gaussian spatial decay. A sympathetic reader would care because the approach supplies an intrinsic diagnostic for topology that does not require external probes or perturbations. The work further shows that this structure remains robust against disorder and contrasts with the exponential decay found in trivial phases.

Core claim

We develop a general framework that links topological phases in driven-dissipative systems to bosonic correlations via the singular value decomposition. Non-Hermitian topology in quadratic Liouvillians is directly encoded in steady-state correlations, providing an intrinsic characterization of topology without external probes. Topological amplification induces disorder-robust long-range order in steady-state correlations at fixed frequency. We introduce a vector-valued topological invariant that captures the total number of singular-value gap closings across the frequency axis. The spatial structure of equal-time correlations encodes global topological information, manifested as Gaussian sp

What carries the argument

Singular value decomposition applied to quadratic Liouvillians, which defines a vector-valued topological invariant by counting singular-value gap closings along the frequency axis and directly shapes the spatial form of steady-state bosonic correlations.

If this is right

  • Topological phases produce long-range order in frequency-resolved steady-state correlations that resists disorder.
  • Equal-time correlations decay Gaussianly with distance inside topological regimes but exponentially inside trivial regimes.
  • Frequency-resolved correlations function as direct signatures of non-Hermitian topological phases.
  • The vector-valued invariant classifies quadratic Liouvillians by extending adiabatic deformation to dissipative systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Correlation measurements alone could suffice to detect topology in quantum simulators such as trapped-ion or circuit platforms.
  • The same decay signatures might appear in fermionic or spin-based driven-dissipative lattices.
  • Robust long-range correlations engineered via topology could improve signal-to-noise ratios in quantum sensing tasks.
  • Extensions to higher-dimensional lattices or weakly interacting regimes could be tested by the same SVD-based diagnostic.

Load-bearing premise

The spatial structure of equal-time correlations encodes global topological information, appearing specifically as Gaussian decay with distance in the topological phase versus exponential decay in trivial phases.

What would settle it

Measure the spatial decay profile of equal-time correlations in a driven-dissipative bosonic chain and check whether the functional form switches from exponential to Gaussian exactly when parameters cross a singular-value gap closing that defines the topological transition.

Figures

Figures reproduced from arXiv: 2605.18394 by Diego Porras, Miguel Clavero Rubio, Tom\'as Ramos.

Figure 1
Figure 1. Figure 1: Schematic representation of normalized frequency-resolved (N¯ ij ) and equal-time (N¯ ij ) correlations in topological and trivial regimes. In the topological phase, frequency-resolved correlations exhibit long-range order, while equal-time correlations display a Gaussian decay. In the trivial regime, both correlations decay exponentially with distance. These results establish two-point correlations as a k… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the dissipative bosonic Kitaev chain. Each site has onsite energy ∆, single-site parametric term gs, and local losses γ. Neighboring sites are connected by non-reciprocal complex hopping Je±iϕ (blue arrows), and parametric couplings gc (red arrows). Without loss of generality, and throughout this work, we focus on Model I in a highly symmetric parameter regime defined by J = gs … view at source ↗
Figure 3
Figure 3. Figure 3: Top: Dimerized version of the dissipative bosonic Kitaev chain. Even (blue) sites correspond to the main chain, characterized by onsite energy ∆, non-reciprocal complex hopping Je±iϕ (blue arrows), local (gs) and nearest-neighbor (gc) parametric couplings (red arrows), and local losses γ. Odd (red) sites represent auxiliary modes, coupled to the main chain via g ′ c , and subject to much stronger dissipati… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Model I: Topological phase diagram for the symmetric parameter set defined in Sec. 4.1. (b) Model II: Topological phase diagram for with gs/g = gc/g = 0.1, g ′ c/g = 3 and γ ′/g = 30 (corresponding to P/g = 3 5 ). (c) Singular value spectrum under OBC in a nontrivial region of (a) with ν = 1, for γ/g = 4 and ω/g = 0 in (a), exhibiting a unique zero singular value separated from the bulk by a gap ∆sg. (… view at source ↗
Figure 5
Figure 5. Figure 5: Model I: Singular values as a function of the frequency ω for PBC. (a) Topologically non-trivial phase with V⃗a = (0, 1, 0, 1, 0), for γ/g = 1.6. (b) Topologically non-trivial phase with V⃗ b = (0, 1, 0), for γ/g = 4. (c) Critical point with V⃗c = (0, 0), for γc/g = 6. (d) Trivial phase with V⃗ d = (0), for γ/g = 8. Each component of V⃗ indicate the topological winding numbers associated with distinct freq… view at source ↗
Figure 6
Figure 6. Figure 6: Model II: Singular values as a function of the frequency ω for PBC. (a) Trivial phase with V⃗a = (0), for g ′ c/g = 2, γ ′/g = 20 (corresponfing to P/g = 2 5 ). (b) Topologically non-trivial phase with V⃗ b = (0, 1, 0), for g ′ c/g = 2.5, γ ′/g = 25 (P/g = 0.5). (c) Topologically non-trivial phase with V⃗c = (0, 1, 2, 1, 0), for g ′ c/g = 3, γ ′/g = 30 (P/g = 3 5 ). (d) Topologically non-trivial phase with… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of topological equivalence under smooth deformations in closed-system Hamiltonians and quadratic Liouvillians of open quantum systems. In closed systems, smooth deformations of topologically protected Hamiltonians preserve the spectral gap and are characterized by a nontrivial winding number, ensuring the existence of zero-energy topological states. In quadratic Liouvillians, smooth deformations… view at source ↗
Figure 8
Figure 8. Figure 8: Normalized frequency-resolved correlations as a function of the site index j, for a fixed reference site i = 10. Colored/black curves indicate topological/trivial phases. The black dashed line marks the critical point separating both regimes. (a, b) Model I, in the symmetric parameter regime defined in Sec. 4.1, with N = 100, and γ/g = 5. (c, d) Model II in the parameter regime defined in Sec. 4.2, with N … view at source ↗
Figure 9
Figure 9. Figure 9: LRO parameter as a function of the frequency ω, for normal (ΛN , solid black) and anomalous (ΛM, dashed red) frequency-resolved correlations. In Model I (panel a), the LRO shows a one-to-one correspondence with the topological phase ω/g ∈ [−1.3, 1.3]. A similar behavior is observed in Model II (panel c), although the topological phase with ν = 2 (ω/g ∈ [−1.28, 1.28]) is indistinguishable from ν = 1 (ω/g ∈ … view at source ↗
Figure 10
Figure 10. Figure 10: Averaged LRO parameter with different disorder strengths W at the symmetric parameter regime of Model I, with γ/g = 4.6, 4.8, 5, 5.2, 5.4, 5.6 (values increasing from blue to magenta). The system size is N = 100 and the number of disorder realizations nr = 500. Plots (a) and (b) show the dependence with disorder and the scaling with the singular value gap, respectively. To characterize the gap closure, we… view at source ↗
Figure 11
Figure 11. Figure 11: Average over disorder realizations of r-parameter at the symmetric point of Model I, and values γ/g = 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8, from blue to magenta. The system size is , N = 100, and the number of disorder realizations nr = 500. Plots (a) and (b) show the dependence with disorder and the scaling with the singular value gap, respectively. We also investigate the spatial decay of normalized freque… view at source ↗
Figure 12
Figure 12. Figure 12: Normalized correlations in frequency space at ω = 0 for different values of the disorder strength at the symmetric regime of Model I, for γ/g = 5.5. The system size is N = 100, and the number of disorder realizations nr = 500. Continuous lines corresponds to exponential fits. To gain further insight, in the limit of small noise W ≪ Wc, we can treat the weak disorder as a perturbation of the bare Green’s f… view at source ↗
Figure 13
Figure 13. Figure 13: Average over disorder realizations of r-parameter as a function of the local dissipation γ/g for different values of weak noise W/g = 0.00, 0.25, 0.50, 0.75. The points represent the results with noise obtained after averaging over 1000 iterations. The solid lines represent the results without noise for the renormalized effective parameters in Eq. (48). For weak disorder, the renormalized theoretical appr… view at source ↗
Figure 14
Figure 14. Figure 14: Model I: Normalized equal-time normal (a), and anomalous (b) correlations centered at i = 24 as a function of the distance j for various dissipation values γ/g. The solid black line shows the theoretical prediction of Eq. (56). The critical value γc/g = 6 (dashed black line) represents the interface between topological (V⃗1 = (0, 1, 0)) and trivial phase (V⃗0 = (0)). The parameters correspond to the symme… view at source ↗
Figure 15
Figure 15. Figure 15: Model II: Normalized equal-time normal (a), and anomalous (b) correlations centered at i = 24 as a function of the distance j for various dissipation values γ/g. The critical value γc1 /g = 5.4 (green dashed line) represents the interface between topological (V⃗1 = (0, 1, 0)) and trivial phase (V⃗0 = (0)). There is another critical value between two topological phases from (V⃗2 = (0, 1, 2, 1, 0)) to (V⃗1 … view at source ↗
Figure 16
Figure 16. Figure 16: LRO parameter as a function of the dissipation rate γ/g for normal (ΛN , solid black line) and anomalous (ΛM, dashed red line) equal-time correlations. (a) Model I: a critical point at γc/g = 6 signals the transition from the topological phase V⃗1 = (0, 1, 0) to the trivial phase V⃗0 = (0), for the symmetric parameter set defined in Sec. 4.1. (b) Model II: two critical point are observed, at γc1 /g = 4.2,… view at source ↗
Figure 17
Figure 17. Figure 17: Edge-state solutions for V± (top) and U± (bottom) lying within the boundaries defined by the ellipses in Eq. (101). Orange and blue shading indicate states localized on the left and right edges, respectively. The remaining parameters correspond to the symmetric parameter regime defined in Sec. 4.1. Using the expression above this quantity can be rewritten as S0 = 2  ω + i γ − g 2  ⟨U0|V0⟩ − 2⟨U0|J + iK … view at source ↗
Figure 18
Figure 18. Figure 18: Comparison between numerical values (dots) and analytical expressions (dashed lines) of the real and imaginary part of the singular vectors Vl0 and Ul0 for different values of the frequency ω/g and γ/g = 5. The remaining parameters correspond to the symmetric parameter regime defined in Sec. 4.1 [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Numerical values (dots) and analytical expressions (solid black lines) of the zero singular values for different values of the frequency ω/g. The green shaded region corresponds to the stable regime for N = 10. Parameter set defined in Sec. 4.1, corresponding to Model I [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
read the original abstract

Driven-dissipative quantum systems can exhibit robust transport and amplification in topological regimes, yet the connection between topology and the extent of correlations remains largely unexplored. In this work, we develop a general framework that links topological phases in driven-dissipative systems to bosonic correlations via the singular value decomposition (SVD). In essence, we claim that non-Hermitian topology in quadratic Liouvillians is directly encoded in steady-state correlations, providing an intrinsic characterization of topology without external probes. We show that topological amplification induces disorder-robust long-range order (LRO) in steady-state correlations at fixed frequency, establishing frequency-resolved correlations as direct signatures of non-Hermitian topological phases. We introduce a vector-valued topological invariant that captures the total number of singular-value gap closings across the frequency axis, extending the concept of adiabatic deformation from topological insulators to the case of topological phases of quadratic Liouvillians. Within this framework, we further demonstrate that the spatial structure of equal-time correlations encodes global topological information, manifested as a Gaussian spatial decay with distance in the topological phase, in contrast to the exponential decay characteristic of trivial phases. These findings open new avenues for quantum sensing and correlation engineering in non-Hermitian systems, with feasible implementations in platforms such as trapped ions and superconducting circuits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a general framework linking non-Hermitian topology in quadratic Liouvillians of driven-dissipative bosonic chains to steady-state bosonic correlations via the singular value decomposition (SVD). It claims that this topology is intrinsically encoded in the correlations, yielding disorder-robust long-range order at fixed frequency in topological phases. A vector-valued topological invariant is introduced that counts the total number of singular-value gap closings across the frequency axis. The paper further asserts that the spatial structure of equal-time correlations encodes global topological information, appearing as Gaussian decay with distance in the topological phase and exponential decay in trivial phases.

Significance. If the central claims are established with explicit derivations, the work would provide an intrinsic, probe-free characterization of non-Hermitian topological phases through steady-state correlations. This could enable new approaches to quantum sensing and correlation engineering in platforms such as trapped ions and superconducting circuits, while extending adiabatic-deformation ideas from closed topological insulators to open quadratic Liouvillians.

major comments (1)
  1. [Abstract and associated derivation sections] The central claim that equal-time correlations exhibit Gaussian spatial decay in the topological phase (contrasted with exponential decay in trivial phases) is not supported by a general derivation. Equal-time correlations are obtained by integrating frequency-resolved correlators over all frequencies; without an explicit analytic step (e.g., contour integration or residue analysis tied to the singular-value gap closings identified by the SVD), the specific Gaussian functional form is not guaranteed by the topology alone and could arise from other spectral features.
minor comments (2)
  1. Clarify the precise definition and normalization of the vector-valued topological invariant when it is first introduced.
  2. Ensure consistent notation for the Liouvillian and its SVD decomposition across all sections and figures.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments, which have helped us identify areas for clarification in the manuscript. We address the major comment point by point below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and associated derivation sections] The central claim that equal-time correlations exhibit Gaussian spatial decay in the topological phase (contrasted with exponential decay in trivial phases) is not supported by a general derivation. Equal-time correlations are obtained by integrating frequency-resolved correlators over all frequencies; without an explicit analytic step (e.g., contour integration or residue analysis tied to the singular-value gap closings identified by the SVD), the specific Gaussian functional form is not guaranteed by the topology alone and could arise from other spectral features.

    Authors: We appreciate the referee highlighting the need for an explicit link between the SVD-based topological invariant and the functional form of the equal-time correlations. In our framework, the frequency-resolved correlators are constructed directly from the SVD of the quadratic Liouvillian, with the vector-valued invariant counting the net number of singular-value gap closings along the frequency axis. The equal-time correlator is the integral of these over all frequencies. While we connect the long-range order at fixed frequency to the topology, we agree that the manuscript does not contain a fully explicit contour-integration argument showing why the integrated spatial decay is Gaussian (rather than, e.g., power-law or exponential) precisely when gap closings are present. Such an analysis would involve deforming the integration contour around the branch cuts or poles whose locations are fixed by the SVD gaps, yielding a Gaussian envelope from the saddle-point contribution in the topological phase and exponential decay when the gaps remain open. We will add this derivation as a new subsection (or appendix) in the revised manuscript, including the residue analysis tied to the SVD and a direct comparison between topological and trivial cases. This will make the claim rigorous and address the possibility that other spectral features could produce the same decay. revision: yes

Circularity Check

0 steps flagged

Framework links topology to correlations via SVD without evident reduction to inputs by construction

full rationale

The paper develops a general framework connecting non-Hermitian topology in quadratic Liouvillians to bosonic steady-state correlations using singular value decomposition and a vector-valued invariant for singular-value gap closings. Claims of Gaussian spatial decay in equal-time correlations for topological phases (versus exponential in trivial phases) are presented as demonstrations within this structure, derived from integrating frequency-resolved correlators. No quoted steps reduce predictions to fitted parameters, self-definitions, or unverified self-citations by construction. The derivation chain remains self-contained with independent content from the SVD-based mapping and adiabatic deformation extension, consistent with external benchmarks for such models.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that quadratic Liouvillians adequately describe the driven-dissipative bosonic chains and on the newly introduced vector-valued topological invariant without independent falsifiable evidence supplied in the abstract.

axioms (1)
  • domain assumption The driven-dissipative bosonic chains are described by quadratic Liouvillians
    This underpins the entire framework for connecting topology to correlations as stated in the abstract.
invented entities (1)
  • vector-valued topological invariant no independent evidence
    purpose: Captures the total number of singular-value gap closings across the frequency axis and extends adiabatic deformation to quadratic Liouvillians
    Newly introduced in the framework; no independent evidence such as a predicted observable outside the paper is mentioned.

pith-pipeline@v0.9.0 · 5770 in / 1315 out tokens · 48677 ms · 2026-05-20T11:04:30.687075+00:00 · methodology

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